Module Identifier MA31210
Module Title NONLINEAR DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS
Co-ordinator Dr Robert J Douglas
Semester Semester 2
Course delivery Lecture   19 x 1 hour lectures
Seminars / Tutorials   3 x 1 hour example classes
Assessment
Assessment TypeAssessment Length/DetailsProportion
Semester Exam2 Hours (written examination)  100%
Supplementary Assessment2 Hours (written examination)  100%

#### Learning outcomes

On completion of this module, a student should be able to:
• interpret conditions for the existence and uniqueness of solutions of autonomous ordinary differential equations;
• explain what is meant by the invariant intervals for an equation;
• classify the critical points of one-dimensional systems;
• classify the critical points of linear two-dimensional systems;
• locate and classify the critical points of two-dimensional nonlinear systems;
• sketch possible phase portraits of two-dimensional nonlinear systems;
• describe simple ecological models and draw appropriate conclusions;
• show how the possibility of limit cycles may be excluded under certain circumstances;
• solve second order systems by matched asymptotic expansions.

#### Brief description

A wide variety of phenomena can be modelled by means of ordinary differential equations.
Very few such equations can be solved explicitly, and in the quantitative theory of differential equations methods have been developed to determine the behaviour of solutions directly from the equation itself. The subject was pioneered in the early part of the twentieth century by Poincare and then by Liapunov. This module provides a thorough grounding in the theory of dynamical systems and nonlinear differential equations.

#### Aims

To provide an introduction to the qualitative theory of nonlinear differential equations, with particular emphasis on the construction of phase portraits of two-dimensional systems and applications.

#### Content

1. Existence and uniqueness of solutions; autonomous and non-autonomous systems.
2. One-dimensional systems; stability and invariance of solutions.
3. Two-dimensional linear systems: classification of critical points.
4. Critical points of two-dimensional nonlinear systems. Increased stability, Poincare-Benedixson theorem an its consequences; construction of possible phase portraits.
5. Modelling by means of two-dimensional nonlinear systems, eg predator-prey models, infectious disease models.
6. Singular pertubations and matched asymptotic expansions. Law of mass action in chemical reactions.